X₈₁₀

CROSSPOINT OF X₁ AND X₁₆₃

X₈₁₀ is the crosspoint of X₁ and X₁₆₃.

5, 6, 7 3, 4, 5 Kepler Kimberling

X₈₁₀ (CROSSPOINT OF X₁ AND X₁₆₃) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {521, 656}, {661, 663}

• Related Centers: X₁ (INCENTER), X₁₆₃ (TRILINEAR PRODUCT X₆*X₁₁₀), X₅₂₁ (ISOGONAL CONJUGATE OF X₁₀₈), X₆₅₆ (CROSSDIFFERENCE OF X₁ AND X₁₉), X₆₆₁ (CROSSDIFFERENCE OF X₁ AND X₂₁), X₆₆₃ (CROSSDIFFERENCE OF X₂ AND X₇)

Trilinears: \(\sin 2A (\cos^{2} B - \cos^{2} C) : :\)

Barycentrics: \(\sin^{2} A \cos A (\cos^{2} B - \cos^{2} C) : :\)

Search Value (6, 9, 13): \(-2.450114674422\)

▸ Alternate Coordinates3 forms

• Trilinears: \(\sin 2A (\sin^{2} B - \sin^{2} C) : :\)

• Trilinears: \(a^{2}(b^{2} - c^{2})(b^{2} + c^{2} - a^{2}) : :\)

• Trilinears: \(\tan B \cot C - \tan C \cot B : :\)

▸ Incident Central Lines3 lines

{521, 656}, {661, 663}, {667, 788}

▸ Functional Relations10 relations

• X₈₁₀ = isogonal conjugate of X₈₁₁

• X₈₁₀ = complement of X₂₁₃₀₀

• X₈₁₀ = anticomplement of X₂₁₂₅₉

• X₈₁₀ = crosspoint of X₁ and X₁₆₃

• X₈₁₀ = crosssum of X₁₆₂ and X₆₆₂

• X₈₁₀ = crossdifference of every pair of points on line X₁₉X₂₇

• X₈₁₀ = X₉₂-isoconjugate of X₆₆₂

• X₈₁₀ = trilinear product of PU(109)

• X₈₁₀ = trilinear product of Jerabek hyperbola intercepts of Lemoine axis

• X₈₁₀ = barycentric product of Jerabek hyperbola intercepts of antiorthic axis

Notes

Let V = X₅₂₁X₆₅₆ = isotomic conjugate of polar conjugate of antiorthic axis and let W = X₅₁₄X₆₆₁ = polar conjugate of isotomic conjugate of antiorthic axis; then X(810 = V∩W. (Randy Hutson, December 26, 2015)